稀疏表达理论作为信号处理领域中诞生的全新理论，由于其巨大的应用潜力，越来越多地引起了相关领域专家和学者的关注。将稀疏表达理论与实际问题相结合，具有重要的研究价值。本文针对稀疏表达理论、方法以及基于稀疏表达的图像处理应用问题，重点围绕稀疏表达算法的改进、稀疏表达在遥感影像融合领域的应用、稀疏表达在天文数据压缩领域的应用等内容进行了探索性的研究。论文首先阐述了稀疏表达的概念与模型、稀疏性度量以及稀疏表达基本算法，分析与比较了常用的过完备字典的构造方法及其性能差异，总结了几种自适应迭代终止条件的优缺点。其次，针对正则化正交匹配追踪(ROMP)算法容易导致部分原子错选的问题，提出了基于回溯追踪的正则化正交匹配追踪(BtROMP)算法。该算法引入回溯策略，在ROMP算法基础上增加回溯追踪过程，以剔除错选的原子，使最终入选的原子能够更加精确地重构原信号。实验结果表明，BtROMP比ROMP算法重构图像的峰值信噪比提高近3dB，最小均方误差明显降低。然后，针对遥感影像融合中存在的问题和发挥稀疏表达理论的优势，本文提出一种基于稀疏表达的遥感影像融合方法。该方法首先通过滑窗方法将原图像分成许多图像块，然后进行稀疏分解，并将全色(Pan)图像块首次迭代得到的稀疏系数值置为0，从而去除Pan图像低频信息，保留高频信息。最后将表示Pan图像高频信息的稀疏系数与表示多光谱(MS)图像各波段的稀疏系数通过线性加权求平均准则融合并重构，得到融合图像。实验结果表明，该方法融合图像在目视效果和定量评价方面均优于其它算法。最后，结合天文数据中背景信息丰富且相邻帧图像相关性大的特点，本文提出一种基于稀疏表达的天文数据压缩方案。该方案首先通过K-Means算法将天文数据中所有帧进行聚类，选择若干“关键帧”，用来训练KSVD字典。再用KSVD字典对各天文图像帧进行稀疏表达，并用改进的游程长度编码算法对稀疏系数矩阵进行编码，输出为二进制文件。本方案编码过程采用的是无损压缩编码方法，仅在稀疏迭代过程中有少量失真。实验结果表明，本方案在对天文数据高保真的情况下，压缩比可达到5:1。

As a new theory in the field of signal processing, sparse representation theory has drawn more and more attentions of researchers in related fields, because of its great potential of application. It is a significant research which combines sparse representation theory and practical problems. The thesis concentrates on the image sparse representation theory, methods and the image processing applications based on sparse representation, making deeply studing for the improvement of sparse representation algorithm, application of sparse representation in the field of remote sensing image fusion and application of sparse representation in the field of astronomical data compression.Firstly, we review the concept and model of sparse representation, measure of sparsity, and the basic methods of sparse representation. And then, we analyze the performance of several commonly used over complete dictionaries and summarize the advantages and disadvantages of several adaptive iteration termination condition.Secondly, to deal with the problem that the ROMP method is easy to cause bad selection of some atoms, a regularized orthogonal matching pursuit algorithm based on backtracking, called BtROMP, is proposed in this paper. The BtROMP utilizes backtracking strategy to get rid of bad atoms in ROMP algorithm, so that it can reconstruct the original signal more accurately. Experiment results show that, the peak signal to noise ratio (PSNR) of BtROMP algorithm increases about 3dB over ROMP algorithm and the mean square error (MSE) is lower.Thirdly, considering the existing problems in remote sensing image fusion and the advantages of sparse representation theory, the remote sensing image fusion method based on sparse representation is put forward in this paper. First of all, the original image is divided into many blocks by sliding window method. Then sparse representation is carried out, with sparse coefficient values set to 0 which is obtained by the first iteration of the Pan image blocks, so as to remove the low frequency information of Pan image. Then, fusing the sparse coefficients of Pan image and MS image. Lastly, we get the fusion image by reconstructing the fusion coefficients. The experimental results show that, the method of fusion image is better than other algorithms in visual effects and quantitative evaluation.Finally, to improve the compression ration of astronomical data, an astronomical data compression program based on sparse representation is presented in the paper. The program first clusters all frames of astronomical data by K-Means algorithm and selects some "key frames" to train KSVD dictionary. And then, we represent all frames of astronomical data by KSVD dictionary, with sparse coefficient matrix encoded by the improved run length coding algorithm. The program uses lossless coding method in the coding process, with only a small amount of distortion in the sparse representation process. The experimental results show that, when astronomical data is not distortion, compression ratio of this program can reach 5:1.